   Chapter 2.8, Problem 54E

Chapter
Section
Textbook Problem

Use the definition of a derivative to find f'(x) and f"(x). Then graph f, f', and f" on a common screen and check to see if your answers are reasonable.f(x) = x3 – 3x

To determine

To find: The first and second derivative of the function f(x)=x33x and sketch its graphs.

Explanation

Given:

The function is, f(x)=x33x.

Formula used:

The derivative of a function f , denoted by f(x), is

f(x)=limh0f(x+h)f(x)h (1)

Calculation:

Obtain the first derivative of the function f(x).

Compute f(x) by using the equation (1),

f(x)=limh0f(x+h)f(x)h=limh0(x+h)33(x+h){x33x}h=limh0x3+h3+3x2h+3xh23x3hx3+3xh

Simplify the numerator and obtain the derivative of the function,

f(x)=limh0h3+3x2h+3xh23hh=limh0h(h2+3x2+3xh3)h

Since the limit h approaches zero but is not equal to zero, cancel the common term h from both the numerator and the denominator,

f(x)=limh0(h2+3x2+3xh3)=02+3x2+3x(0)3=3x23

Thus, the first derivative of the function is, f(x)=3x23.

Obtain the second derivative of the function f(x)

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